\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  -2.0 & +  1.00 x_{1} & +  4.00 x_{2} & -1.00 x_{3} & +  2.00 x_{4} & +  3.00 x_{5} & +  1.00 x_{6} & +  9.00 x_{7}\\
 x_{9}   &  3.0 & -9.00 x_{1} & -5.00 x_{2} & -10.00 x_{3} & -1.00 x_{4} & +  2.00 x_{5} & -6.00 x_{6} & +  7.00 x_{7}\\
 x_{10}   &  -3.0 & +  5.00 x_{1} & + 10.00 x_{2} & +  1.00 x_{3} & +  4.00 x_{4} & -3.00 x_{5} & -9.00 x_{6} & -9.00 x_{7}\\
 x_{11}   &  0.0 & -3.00 x_{1} & +  6.00 x_{2} & -5.00 x_{3} & -9.00 x_{4} & +  5.00 x_{5} & -2.00 x_{6} & -3.00 x_{7}\\
 x_{12}   &  -2.0 & -2.00 x_{1} & + 10.00 x_{2} & -6.00 x_{3} & -6.00 x_{4} &   & +  4.00 x_{6} & + 10.00 x_{7}\\
 x_{13}   &  0.0 & +  6.00 x_{1} &   & +  5.00 x_{3} & -9.00 x_{4} & +  2.00 x_{5} & -1.00 x_{6} & -7.00 x_{7}\\
\hline
z    &  0.0 & +  3.00 x_{1} &   & -1.00 x_{3} & -1.00 x_{4} & +  4.00 x_{5} & -4.00 x_{6} & +  1.00 x_{7}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -1.00 y_{8} & +  9.00 y_{9} & -5.00 y_{10} & +  3.00 y_{11} & +  2.00 y_{12} & -6.00 y_{13}\\
 y_{2}   &  1.0 & -4.00 y_{8} & +  5.00 y_{9} & -10.00 y_{10} & -6.00 y_{11} & -10.00 y_{12} &   \\
 y_{3}   &  1.0 & +  1.00 y_{8} & + 10.00 y_{9} & -1.00 y_{10} & +  5.00 y_{11} & +  6.00 y_{12} & -5.00 y_{13}\\
 y_{4}   &  1.0 & -2.00 y_{8} & +  1.00 y_{9} & -4.00 y_{10} & +  9.00 y_{11} & +  6.00 y_{12} & +  9.00 y_{13}\\
 y_{5}   &  1.0 & -3.00 y_{8} & -2.00 y_{9} & +  3.00 y_{10} & -5.00 y_{11} &   & -2.00 y_{13}\\
 y_{6}   &  1.0 & -1.00 y_{8} & +  6.00 y_{9} & +  9.00 y_{10} & +  2.00 y_{11} & -4.00 y_{12} & +  1.00 y_{13}\\
 y_{7}   &  1.0 & -9.00 y_{8} & -7.00 y_{9} & +  9.00 y_{10} & +  3.00 y_{11} & -10.00 y_{12} & +  7.00 y_{13}\\
\hline
z    &  -0 & +  2.00 y_{8} & -3.00 y_{9} & +  3.00 y_{10} &   & +  2.00 y_{12} &   \\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{13}   &  0.119533527697 & -0.01 y_{7} & +  1.16 y_{9} & +  0.08 y_{2} & +  1.07 y_{11} & +  1.01 y_{12} & -0.18 y_{1}\\
 y_{10}   &  0.0131195335277 & +  0.03 y_{7} & +  0.32 y_{9} & -0.09 y_{2} & -0.76 y_{11} & -0.62 y_{12} & +  0.04 y_{1}\\
 y_{3}   &  0.606413994169 & -0.05 y_{7} & +  4.34 y_{9} & -0.33 y_{2} & +  0.83 y_{11} & +  0.63 y_{12} & +  0.78 y_{1}\\
 y_{4}   &  1.5889212828 & -0.10 y_{7} & +  9.24 y_{9} & +  1.12 y_{2} & + 20.84 y_{11} & + 19.46 y_{12} & -1.61 y_{1}\\
 y_{5}   &  0.148688046647 & +  0.40 y_{7} & -4.68 y_{9} & -0.34 y_{2} & -10.62 y_{11} & -1.04 y_{12} & +  0.80 y_{1}\\
 y_{6}   &  1.02040816327 & +  0.39 y_{7} & +  9.61 y_{9} & -0.69 y_{2} & -4.18 y_{11} & -7.63 y_{12} & +  0.29 y_{1}\\
 y_{8}   &  0.217201166181 & -0.09 y_{7} & +  0.44 y_{9} & -0.03 y_{2} & +  0.40 y_{11} & -0.95 y_{12} & -0.10 y_{1}\\
\hline
z    &  0.473760932945 & -0.07 y_{7} & -1.14 y_{9} & -0.32 y_{2} & -1.48 y_{11} & -1.76 y_{12} & -0.08 y_{1}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.069970845481 & +  0.01 x_{13} & -0.03 x_{10} & +  0.05 x_{3} & +  0.10 x_{4} & -0.40 x_{5} & -0.39 x_{6} & +  0.09 x_{8}\\
 x_{9}   &  1.1443148688 & -1.16 x_{13} & -0.32 x_{10} & -4.34 x_{3} & -9.24 x_{4} & +  4.68 x_{5} & -9.61 x_{6} & -0.44 x_{8}\\
 x_{2}   &  0.322157434402 & -0.08 x_{13} & +  0.09 x_{10} & +  0.33 x_{3} & -1.12 x_{4} & +  0.34 x_{5} & +  0.69 x_{6} & +  0.03 x_{8}\\
 x_{11}   &  1.47813411079 & -1.07 x_{13} & +  0.76 x_{10} & -0.83 x_{3} & -20.84 x_{4} & + 10.62 x_{5} & +  4.18 x_{6} & -0.40 x_{8}\\
 x_{12}   &  1.75801749271 & -1.01 x_{13} & +  0.62 x_{10} & -0.63 x_{3} & -19.46 x_{4} & +  1.04 x_{5} & +  7.63 x_{6} & +  0.95 x_{8}\\
 x_{1}   &  0.0816326530612 & +  0.18 x_{13} & -0.04 x_{10} & -0.78 x_{3} & +  1.61 x_{4} & -0.80 x_{5} & -0.29 x_{6} & +  0.10 x_{8}\\
\hline
z    &  -0.473760932945 & -0.12 x_{13} & -0.01 x_{10} & -0.61 x_{3} & -1.59 x_{4} & -0.15 x_{5} & -1.02 x_{6} & -0.22 x_{8}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.069970845481 & +  0.01 x_{13} & -0.03 x_{10} & +  0.05 x_{3} & +  0.10 x_{4} & -0.40 x_{5} & -0.39 x_{6} & +  0.09 x_{8}\\
 x_{9}   &  1.1443148688 & -1.16 x_{13} & -0.32 x_{10} & -4.34 x_{3} & -9.24 x_{4} & +  4.68 x_{5} & -9.61 x_{6} & -0.44 x_{8}\\
 x_{2}   &  0.322157434402 & -0.08 x_{13} & +  0.09 x_{10} & +  0.33 x_{3} & -1.12 x_{4} & +  0.34 x_{5} & +  0.69 x_{6} & +  0.03 x_{8}\\
 x_{11}   &  1.47813411079 & -1.07 x_{13} & +  0.76 x_{10} & -0.83 x_{3} & -20.84 x_{4} & + 10.62 x_{5} & +  4.18 x_{6} & -0.40 x_{8}\\
 x_{12}   &  1.75801749271 & -1.01 x_{13} & +  0.62 x_{10} & -0.63 x_{3} & -19.46 x_{4} & +  1.04 x_{5} & +  7.63 x_{6} & +  0.95 x_{8}\\
 x_{1}   &  0.0816326530612 & +  0.18 x_{13} & -0.04 x_{10} & -0.78 x_{3} & +  1.61 x_{4} & -0.80 x_{5} & -0.29 x_{6} & +  0.10 x_{8}\\
\hline
z    &  0.314868804665 & +  0.57 x_{13} & -0.16 x_{10} & -3.28 x_{3} & +  3.93 x_{4} & +  1.22 x_{5} & -5.24 x_{6} & +  0.39 x_{8}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.069970845481 & +  0.01 x_{13} & -0.03 x_{10} & +  0.05 x_{3} & +  0.10 x_{4} & -0.40 x_{5} & -0.39 x_{6} & +  0.09 x_{8}\\
 x_{9}   &  1.1443148688 & -1.16 x_{13} & -0.32 x_{10} & -4.34 x_{3} & -9.24 x_{4} & +  4.68 x_{5} & -9.61 x_{6} & -0.44 x_{8}\\
 x_{2}   &  0.322157434402 & -0.08 x_{13} & +  0.09 x_{10} & +  0.33 x_{3} & -1.12 x_{4} & +  0.34 x_{5} & +  0.69 x_{6} & +  0.03 x_{8}\\
 x_{11}   &  1.47813411079 & -1.07 x_{13} & +  0.76 x_{10} & -0.83 x_{3} & -20.84 x_{4} & + 10.62 x_{5} & +  4.18 x_{6} & -0.40 x_{8}\\
 x_{12}   &  1.75801749271 & -1.01 x_{13} & +  0.62 x_{10} & -0.63 x_{3} & -19.46 x_{4} & +  1.04 x_{5} & +  7.63 x_{6} & +  0.95 x_{8}\\
 x_{1}   &  0.0816326530612 & +  0.18 x_{13} & -0.04 x_{10} & -0.78 x_{3} & +  1.61 x_{4} & -0.80 x_{5} & -0.29 x_{6} & +  0.10 x_{8}\\
\hline
z    &  0.314868804665 & +  0.57 x_{13} & -0.16 x_{10} & -3.28 x_{3} & +  3.93 x_{4} & +  1.22 x_{5} & -5.24 x_{6} & +  0.39 x_{8}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.0767939571968 & +  0.01 x_{13} & -0.03 x_{10} & +  0.05 x_{3} & -0.00 x_{11} & -0.35 x_{5} & -0.37 x_{6} & +  0.09 x_{8}\\
 x_{9}   &  0.489089383131 & -0.68 x_{13} & -0.66 x_{10} & -3.97 x_{3} & +  0.44 x_{11} & -0.03 x_{5} & -11.47 x_{6} & -0.27 x_{8}\\
 x_{2}   &  0.242761225346 & -0.02 x_{13} & +  0.05 x_{10} & +  0.38 x_{3} & +  0.05 x_{11} & -0.23 x_{5} & +  0.47 x_{6} & +  0.05 x_{8}\\
 x_{4}   &  0.0709190096517 & -0.05 x_{13} & +  0.04 x_{10} & -0.04 x_{3} & -0.05 x_{11} & +  0.51 x_{5} & +  0.20 x_{6} & -0.02 x_{8}\\
 x_{12}   &  0.378094838439 & -0.01 x_{13} & -0.09 x_{10} & +  0.14 x_{3} & +  0.93 x_{11} & -8.88 x_{5} & +  3.73 x_{6} & +  1.32 x_{8}\\
 x_{1}   &  0.19597146454 & +  0.10 x_{13} & +  0.02 x_{10} & -0.84 x_{3} & -0.08 x_{11} & +  0.03 x_{5} & +  0.04 x_{6} & +  0.07 x_{8}\\
\hline
z    &  0.593789341167 & +  0.36 x_{13} & -0.01 x_{10} & -3.43 x_{3} & -0.19 x_{11} & +  3.22 x_{5} & -4.46 x_{6} & +  0.32 x_{8}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.061999873984 & +  0.01 x_{13} & -0.03 x_{10} & +  0.04 x_{3} & -0.04 x_{11} & +  0.04 x_{12} & -0.51 x_{6} & +  0.03 x_{8}\\
 x_{9}   &  0.487933967614 & -0.68 x_{13} & -0.66 x_{10} & -3.97 x_{3} & +  0.44 x_{11} & +  0.00 x_{12} & -11.48 x_{6} & -0.27 x_{8}\\
 x_{2}   &  0.232987839456 & -0.02 x_{13} & +  0.05 x_{10} & +  0.37 x_{3} & +  0.03 x_{11} & +  0.03 x_{12} & +  0.37 x_{6} & +  0.02 x_{8}\\
 x_{4}   &  0.0926217629639 & -0.05 x_{13} & +  0.03 x_{10} & -0.03 x_{3} & +  0.01 x_{11} & -0.06 x_{12} & +  0.41 x_{6} & +  0.06 x_{8}\\
 x_{5}   &  0.0425776573625 & -0.00 x_{13} & -0.01 x_{10} & +  0.02 x_{3} & +  0.11 x_{11} & -0.11 x_{12} & +  0.42 x_{6} & +  0.15 x_{8}\\
 x_{1}   &  0.197073278306 & +  0.10 x_{13} & +  0.02 x_{10} & -0.84 x_{3} & -0.07 x_{11} & -0.00 x_{12} & +  0.05 x_{6} & +  0.07 x_{8}\\
\hline
z    &  0.730908575389 & +  0.36 x_{13} & -0.05 x_{10} & -3.38 x_{3} & +  0.15 x_{11} & -0.36 x_{12} & -3.10 x_{6} & +  0.80 x_{8}\\
\end{array}\]


 $ x_{8} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.123076923077 & -0.08 x_{13} & -0.11 x_{10} & -0.46 x_{3} & +  0.01 x_{11} & +  0.04 x_{12} & -1.95 x_{6} & -0.13 x_{9}\\
 x_{8}   &  1.80512820513 & -2.53 x_{13} & -2.44 x_{10} & -14.68 x_{3} & +  1.63 x_{11} & +  0.01 x_{12} & -42.46 x_{6} & -3.70 x_{9}\\
 x_{2}   &  0.260256410256 & -0.06 x_{13} & +  0.01 x_{10} & +  0.15 x_{3} & +  0.05 x_{11} & +  0.03 x_{12} & -0.27 x_{6} & -0.06 x_{9}\\
 x_{4}   &  0.194871794872 & -0.20 x_{13} & -0.11 x_{10} & -0.86 x_{3} & +  0.10 x_{11} & -0.06 x_{12} & -1.99 x_{6} & -0.21 x_{9}\\
 x_{5}   &  0.311538461538 & -0.38 x_{13} & -0.37 x_{10} & -2.17 x_{3} & +  0.35 x_{11} & -0.11 x_{12} & -5.91 x_{6} & -0.55 x_{9}\\
 x_{1}   &  0.332051282051 & -0.09 x_{13} & -0.16 x_{10} & -1.94 x_{3} & +  0.05 x_{11} & -0.00 x_{12} & -3.13 x_{6} & -0.28 x_{9}\\
\hline
z    &  2.17051282051 & -1.66 x_{13} & -1.99 x_{10} & -15.09 x_{3} & +  1.45 x_{11} & -0.35 x_{12} & -36.97 x_{6} & -2.95 x_{9}\\
\end{array}\]


 $ x_{11} $ enters and Unbounded Dictionary!


 $ x_{11} $ enters and Unbounded Dictionary!
\end{document}
